In the limit \(y\rightarrow 1\) , \(\Omega _\infty (\gamma )\)  reduces to the signed Euler characteristic
\((-1)^{\dim _{{C}}\mathcal {M}_\infty (\gamma )} e(\mathcal {M}_\infty (\gamma ))\) .
When the inequality (REF ) is saturated, \(\mathcal {M}_\infty (\gamma )\) 
has Picard rank \(b_2(\mathcal {M}_\infty (\gamma ))=1\) , or 2 when the inequality is strict.
The Gieseker index for sheaves on \({P}^2\)  with arbitrary rank was determined in [1]}, by relating it to the Gieseker index for sheaves on the Hirzebruch surface
\({F}_1\)  (which coincides with the blow-up of \({P}^2\)  at one point) and using wall-crossing arguments.
Quite remarkably, the generating series of Gieseker indices with fixed rank \(r\)  and degree \(d\) 
are conjectured to transform as mock Jacobi forms of the parameters \((\tau ,w)\)  conjugate to the
second Chern class \(\textnormal {ch}_2\)  and Betti degree \(p\)  in (REF ) [2]}. The simplest case is for
rank 1, where the moduli space \(\mathcal {M}_\infty ([1,0,1-n))\)  coincides with the Hilbert scheme of \(n\)  points on \({P}^{2}\) , and the
generating series is an actual Jacobi form [3]},
\(\begin{aligned}& \sum _{n\ge 0}\Omega _\infty ([1,0,1-n)) \, q^{n} = \frac{Iq^{1/8} (y-1/y)}{\theta _1(q,y^2)} \\& \quad = 1 + (y^2+1+1/y^2) q +(y^2+1+1/y^2)^2 q^2 + (y^6+2y^4+5y^2+6+\dots ) q^3 + \dots \\& \quad \stackrel{y\rightarrow 1}{\rightarrow } 1 + 3 q + 9 q^2 + 22 q^3 + 51 q^4 + 108 q^5 + 221 q^6 + 429 q^7 +810 q^8 + 1479 q^9 + \dots \end{aligned}\) 
